If you give stock to your child, they get the stock with your purchase date and your cost basis. (Assuming you give them something that has gone up in value. If it has gone down, things get complicated. But I don't think that's your issue.)

That's the part Phil mentioned. But there's another issue. The "kiddie" tax.

If your dependent child under the age of 23 has unearned income - and capital gains are unearned income - they will pay tax based on your income tax rate.

In your case, you mention the child is over 18, but has no income. Is the child your dependent? If so, you will save only a minor amount of taxes. The child will get the first $800 (or is it $850 - durn number changing every year - grrr!) tax free, and the next $800/850 at their rate, which is probably 5% for long term capital gains. The balance of the gain will be taxed at your rate.